Interpretation, Prediction and Confidence Intervals
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1 Interpretation, Prediction and Confidence Intervals Merlise Clyde September 15, 2017
2 Last Class Model for log brain weight as a function of log body weight Nested Model Comparison using ANOVA led to model with parallel lines
3 Last Class Model for log brain weight as a function of log body weight Nested Model Comparison using ANOVA led to model with parallel lines Why does model with the 3 indicator variable contain the other models? log(brain) = β 0 +log(body)β 1 +Dino.Tβ 2 +Dino.Bβ 3 +Dino.Dβ 4 +ɛ
4 Check residuals Residuals vs Fitted Normal Q Q Residuals Human Rhesus monkey Chimpanzee Standardized residuals Human Rhesus monkey Chimpanzee Fitted values Theoretical Quantiles Standardized residuals Scale Location Human Rhesus monkey Chimpanzee Standardized residuals Human Cook's distance Residuals vs Leverage Triceratops Brachiosaurus Fitted values Leverage
5 Coefficient Summaries Estimate Std. Error t value Pr(> t ) (Intercept) log(body) DinoTRUE
6 Distribution of Coefficients Joint Distribution under normality ˆβ σ 2 N(β, σ 2 (X T X) 1 )
7 Distribution of Coefficients Joint Distribution under normality ˆβ σ 2 N(β, σ 2 (X T X) 1 ) Distribution of SSE SSE/σ 2 χ 2 (n p)
8 Distribution of Coefficients Joint Distribution under normality ˆβ σ 2 N(β, σ 2 (X T X) 1 ) Distribution of SSE SSE/σ 2 χ 2 (n p) Marginal distribution ˆβ j β j SE( ˆβ j ) St(n p) SE( ˆβ j ) = ˆσ [X T X] 1 ] jj ] ˆσ 2 = SSE n p
9 Confidence Intervals (1 α/2)100% Confidence interval for β j ˆβ j ± t n p,α/2 SE( ˆβ j ) kable(confint(brain2.lm)) 2.5 % 97.5 % (Intercept) log(body) DinoTRUE
10 Converting to Original Units Model after exponentiating brain = e ˆβ 0 +log(body) ˆβ 1 +Dino ˆβ 2 = e ˆβ 0 e log(body) ˆβ 1 e Dino ˆβ 2 = e ˆβ 0 body ˆβ 1 e Dino ˆβ 2
11 Converting to Original Units Model after exponentiating brain = e ˆβ 0 +log(body) ˆβ 1 +Dino ˆβ 2 = e ˆβ 0 e log(body) ˆβ 1 e Dino ˆβ 2 = e ˆβ 0 body ˆβ 1 e Dino ˆβ 2 10% increase in body weight implies a brain 1.10 = e ˆβ 0 (1.10 body)ˆβ 1 e Dino ˆβ 2 = 1.10 ˆβ 1 e ˆβ 0 body ˆβ 1 e Dino ˆβ 2
12 Converting to Original Units Model after exponentiating brain = e ˆβ 0 +log(body) ˆβ 1 +Dino ˆβ 2 = e ˆβ 0 e log(body) ˆβ 1 e Dino ˆβ 2 = e ˆβ 0 body ˆβ 1 e Dino ˆβ 2 10% increase in body weight implies a brain 1.10 = e ˆβ 0 (1.10 body)ˆβ 1 e Dino ˆβ 2 = 1.10 ˆβ 1 e ˆβ 0 body ˆβ 1 e Dino ˆβ ˆβ 1 = or a 7.4% increase in brain weight
13 95% Confidence interval To obtain a 95% confidence interval, (1.10 CI 1) % 97.5 % body
14 Interpretation of Intercept Evalutate model with predictors = 0 log(brain) = ˆβ 0 + log(body) ˆβ 1 + Dino ˆβ 2
15 Interpretation of Intercept Evalutate model with predictors = 0 log(brain) = ˆβ 0 + log(body) ˆβ 1 + Dino ˆβ 2 For a non-dinosaur, if log(body) = 0 (body weight = 1 kilogram), we expect that brain weight will be 2.16 log(grams)???
16 Interpretation of Intercept Evalutate model with predictors = 0 log(brain) = ˆβ 0 + log(body) ˆβ 1 + Dino ˆβ 2 For a non-dinosaur, if log(body) = 0 (body weight = 1 kilogram), we expect that brain weight will be 2.16 log(grams)??? Exponentiate: predicted brain weight for non-dinosaur with a 1 kg body weight is e ˆβ 0 = 8.69 grams
17 Plot of Fitted Values library(ggplot2) beta= coef(brain2.lm) gp = ggplot(animals, aes(y=log(brain), x=log(body))) + geom_point(aes(colour=factor(dino))) + geom_abline(aes(intercept=beta[1], slope=beta[2])) + geom_abline(aes(intercept=(beta[1]+beta[3]), slope=beta[2]))
18 Plot of Fitted Values 8 6 log(brain) 4 factor(dino) FALSE TRUE log(body)
19 Confidence Intervals for the f (x) Point Estimate f (x) = x T ˆβ
20 Confidence Intervals for the f (x) Point Estimate Distribution of MLE given σ f (x) = x T ˆβ f (x) N(f (x), σ 2 x T (X T X) 1 x)
21 Confidence Intervals for the f (x) Point Estimate Distribution of MLE given σ Distribution of pivotal quantity f (x) = x T ˆβ f (x) N(f (x), σ 2 x T (X T X) 1 x) f (x) f (x) t(n p) ˆσ 2 x T (X T X) 1 x
22 Confidence Intervals for the f (x) Point Estimate Distribution of MLE given σ Distribution of pivotal quantity Confidence interval f (x) = x T ˆβ f (x) N(f (x), σ 2 x T (X T X) 1 x) f (x) f (x) t(n p) ˆσ 2 x T (X T X) 1 x f (x) ± t α/2 ˆσ 2 x T (X T X) 1 x
23 Prediction Intervals for Y at x Model Y = x T β + ɛ
24 Prediction Intervals for Y at x Model Y = x T β + ɛ Y independent of other Y s
25 Prediction Intervals for Y at x Model Y independent of other Y s Prediction error Y = x T β + ɛ Y f (x) = x T β f (x ) + ɛ
26 Prediction Intervals for Y at x Model Y independent of other Y s Prediction error Variance Y = x T β + ɛ Y f (x) = x T β f (x ) + ɛ Var(Y f (x)) = Var(x T β f (x )) + Var(ɛ ) = σ 2 x T (X T X) 1 x + σ 2 = σ 2 (1 + x T (X T X) 1 x )
27 Prediction Intervals for Y at x Model Y independent of other Y s Prediction error Variance Y = x T β + ɛ Y f (x) = x T β f (x ) + ɛ Var(Y f (x)) = Var(x T β f (x )) + Var(ɛ ) Prediction Intervals = σ 2 x T (X T X) 1 x + σ 2 = σ 2 (1 + x T (X T X) 1 x ) f (x) ± t α/2 ˆσ 2 (1 + x T (X T X) 1 x )
28 Predictions for 259 gram cockatoo 9 6 log(brain) 3 factor(dino) FALSE TRUE log(body)
29 Predictions in original units 95% Confidence Interval for f (x) newdata = data.frame(body=.0259, Dino=FALSE) fit = predict(brain2.lm, newdata=newdata, interval="confidence", se=t) 95% Prediction Interval for Brain Weight pred = predict(brain2.lm, newdata=newdata, interval="predict", se=t)
30 CI/Predictions in original units for body=259 g 95% Confidence Interval for f (x) exp(fit$fit) ## fit lwr upr ## % Prediction Interval for Brain Weight exp(pred$fit) ## fit lwr upr ## % confident that the brain weight will be between 0.11 and 2.81 grams
31 Summary Linear predictors may be based on functions of other predictors (dummy variables, interactions, non-linear terms)
32 Summary Linear predictors may be based on functions of other predictors (dummy variables, interactions, non-linear terms) need to change back to original units
33 Summary Linear predictors may be based on functions of other predictors (dummy variables, interactions, non-linear terms) need to change back to original units log transform useful for non-negative responses (ensures predictions are non-negative)
34 Summary Linear predictors may be based on functions of other predictors (dummy variables, interactions, non-linear terms) need to change back to original units log transform useful for non-negative responses (ensures predictions are non-negative) Be careful of units of data
35 Summary Linear predictors may be based on functions of other predictors (dummy variables, interactions, non-linear terms) need to change back to original units log transform useful for non-negative responses (ensures predictions are non-negative) Be careful of units of data plots should show units
36 Summary Linear predictors may be based on functions of other predictors (dummy variables, interactions, non-linear terms) need to change back to original units log transform useful for non-negative responses (ensures predictions are non-negative) Be careful of units of data plots should show units summary statements should include units
37 Summary Linear predictors may be based on functions of other predictors (dummy variables, interactions, non-linear terms) need to change back to original units log transform useful for non-negative responses (ensures predictions are non-negative) Be careful of units of data plots should show units summary statements should include units Goodness of fit measure: R 2 and Adjusted R 2 depend on scale R 2 is percent variation in Y that is explained by the model where SST = i (Y i Ȳ ) 2 R 2 = 1 SSE/SST
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